Monoids add an identity element, and homomorphisms describe structure-preserving maps between monoids.
Monoid
A monoid is a semigroup with an identity element e such that e * a = a and
a * e = a for all a in the set.
Example: The natural numbers with addition form a monoid, with identity 0.
Homomorphism
Given monoids (M, *) and (N, o), a homomorphism f : M -> N preserves the
operation: f(a * b) = f(a) o f(b) for all a, b in M.
Homomorphisms let us compare algebraic structures while respecting their operations.